List of integrals of trigonometric functions

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The following is a list of integrals (antiderivative functions) of trigonometric functions. For antiderivatives involving both exponential and trigonometric functions, see List of integrals of exponential functions. For a complete list of antiderivative functions, see Lists of integrals. For the special antiderivatives involving trigonometric functions, see Trigonometric integral.[1]

Generally, if the function sin⁡x is any trigonometric function, and cos⁡x is its derivative,

∫acos⁡nxdx=ansin⁡nx+C

In all formulas the constant a is assumed to be nonzero, and C denotes the constant of integration.

## Contents

* 1 Integrands involving only sine
* 2 Integrands involving only cosine
* 3 Integrands involving only tangent
* 4 Integrands involving only secant
* 5 Integrands involving only cosecant
* 6 Integrands involving only cotangent
* 7 Integrands involving both sine and cosine
* 8 Integrands involving both sine and tangent
* 9 Integrand involving both cosine and tangent
* 10 Integrand involving both sine and cotangent
* 11 Integrand involving both cosine and cotangent
* 12 Integrand involving both secant and tangent
* 13 Integrand involving both cosecant and cotangent
* 14 Integrals in a quarter period
* 15 Integrals with symmetric limits
* 16 Integral over a full circle
* 17 See also
* 18 References

## Integrands involving only sine

* ∫sin⁡axdx=−1acos⁡ax+C
* ∫sin2axdx=x2−14asin⁡2ax+C=x2−12asin⁡axcos⁡ax+C
* ∫sin3axdx=cos⁡3ax12a−3cos⁡ax4a+C
* ∫xsin2axdx=x24−x4asin⁡2ax−18a2cos⁡2ax+C
* ∫x2sin2axdx=x36−(x24a−18a3)sin⁡2ax−x4a2cos⁡2ax+C
* ∫xsin⁡axdx=sin⁡axa2−xcos⁡axa+C
* ∫(sin⁡b1x)(sin⁡b2x)dx=sin⁡((b2−b1)x)2(b2−b1)−sin⁡((b1+b2)x)2(b1+b2)+C(for |b1|≠|b2|)
* ∫sinnaxdx=−sinn−1axcos⁡axna+n−1n∫sinn−2axdx(for n>0)
* ∫dxsin⁡ax=−1aln⁡|csc⁡ax+cot⁡ax|+C=1aln⁡|tan⁡ax2|+C
* ∫dxsinnax=cos⁡axa(1−n)sinn−1ax+n−2n−1∫dxsinn−2ax(for n>1)
* ∫xnsin⁡axdx=−xnacos⁡ax+na∫xn−1cos⁡axdx=∑k=02k≤n(−1)k+1xn−2ka1+2kn!(n−2k)!cos⁡ax+∑k=02k+1≤n(−1)kxn−1−2ka2+2kn!(n−2k−1)!sin⁡ax=−∑k=0nxn−ka1+kn!(n−k)!cos⁡(ax+kπ2)(for n>0)
* ∫sin⁡axxdx=∑n=0∞(−1)n(ax)2n+1(2n+1)⋅(2n+1)!+C
* ∫sin⁡axxndx=−sin⁡ax(n−1)xn−1+an−1∫cos⁡axxn−1dx
* ∫sin⁡(ax2+bx+c)dx={aπ2cos⁡(b2−4ac4a)S(2ax+b2aπ)+aπ2sin⁡(b2−4ac4a)C(2ax+b2aπ)tob2−4ac>0aπ2cos⁡(b2−4ac4a)S(2ax+b2aπ)−aπ2sin⁡(b2−4ac4a)C(2ax+b2aπ)tob2−4ac<0fora╱=0,a>0
* ∫dx1±sin⁡ax=1atan⁡(ax2∓π4)+C
* ∫xdx1+sin⁡ax=xatan⁡(ax2−π4)+2a2ln⁡|cos⁡(ax2−π4)|+C
* ∫xdx1−sin⁡ax=xacot⁡(π4−ax2)+2a2ln⁡|sin⁡(π4−ax2)|+C
* ∫sin⁡axdx1±sin⁡ax=±x+1atan⁡(π4∓ax2)+C

## Integrands involving only cosine

* ∫cos⁡axdx=1asin⁡ax+C
* ∫cos2axdx=x2+14asin⁡2ax+C=x2+12asin⁡axcos⁡ax+C
* ∫cosnaxdx=cosn−1axsin⁡axna+n−1n∫cosn−2axdx(for n>0)
* ∫xcos⁡axdx=cos⁡axa2+xsin⁡axa+C
* ∫x2cos2axdx=x36+(x24a−18a3)sin⁡2ax+x4a2cos⁡2ax+C
* ∫xncos⁡axdx=xnsin⁡axa−na∫xn−1sin⁡axdx=∑k=02k+1≤n(−1)kxn−2k−1a2+2kn!(n−2k−1)!cos⁡ax+∑k=02k≤n(−1)kxn−2ka1+2kn!(n−2k)!sin⁡ax=∑k=0n(−1)⌊k/2⌋xn−ka1+kn!(n−k)!cos⁡(ax−(−1)k+12π2)=∑k=0nxn−ka1+kn!(n−k)!sin⁡(ax+kπ2)(for n>0)
* ∫cos⁡axxdx=ln⁡|ax|+∑k=1∞(−1)k(ax)2k2k⋅(2k)!+C
* ∫cos⁡axxndx=−cos⁡ax(n−1)xn−1−an−1∫sin⁡axxn−1dx(for n≠1)
* ∫dxcos⁡ax=1aln⁡|tan⁡(ax2+π4)|+C=1aln⁡|sec⁡ax+tan⁡ax|+C
* ∫dxcosnax=sin⁡axa(n−1)cosn−1ax+n−2n−1∫dxcosn−2ax(for n>1)
* ∫dx1+cos⁡ax=1atan⁡ax2+C
* ∫dx1−cos⁡ax=−1acot⁡ax2+C
* ∫xdx1+cos⁡ax=xatan⁡ax2+2a2ln⁡|cos⁡ax2|+C
* ∫xdx1−cos⁡ax=−xacot⁡ax2+2a2ln⁡|sin⁡ax2|+C
* ∫cos⁡axdx1+cos⁡ax=x−1atan⁡ax2+C
* ∫cos⁡axdx1−cos⁡ax=−x−1acot⁡ax2+C
* ∫(cos⁡a1x)(cos⁡a2x)dx=sin⁡((a2−a1)x)2(a2−a1)+sin⁡((a2+a1)x)2(a2+a1)+C(for |a1|≠|a2|)

## Integrands involving only tangent

* ∫tan⁡axdx=−1aln⁡|cos⁡ax|+C=1aln⁡|sec⁡ax|+C
* ∫tan2xdx=tan⁡x−x+C
* ∫tannaxdx=1a(n−1)tann−1ax−∫tann−2axdx(for n≠1)
* ∫dxqtan⁡ax+p=1p2+q2(px+qaln⁡|qsin⁡ax+pcos⁡ax|)+C(for p2+q2≠0)
* ∫dxtan⁡ax±1=±x2+12aln⁡|sin⁡ax±cos⁡ax|+C
* ∫tan⁡axdxtan⁡ax±1=x2∓12aln⁡|sin⁡ax±cos⁡ax|+C

## Integrands involving only secant

* ∫sec⁡axdx=1aln⁡|sec⁡ax+tan⁡ax|+C=1aln⁡|tan⁡(ax2+π4)|+C=1aartanh⁡(sin⁡ax)+C
* ∫sec2xdx=tan⁡x+C
* ∫sec3xdx=12sec⁡xtan⁡x+12ln⁡|sec⁡x+tan⁡x|+C.
* ∫secnaxdx=secn−2axtan⁡axa(n−1)+n−2n−1∫secn−2axdx (for n≠1)
* ∫dxsec⁡x+1=x−tan⁡x2+C
* ∫dxsec⁡x−1=−x−cot⁡x2+C

## Integrands involving only cosecant

* ∫csc⁡axdx=−1aln⁡|csc⁡ax+cot⁡ax|+C=1aln⁡|csc⁡ax−cot⁡ax|+C=1aln⁡|tan⁡(ax2)|+C
* ∫csc2xdx=−cot⁡x+C
* ∫csc3xdx=−12csc⁡xcot⁡x−12ln⁡|csc⁡x+cot⁡x|+C=−12csc⁡xcot⁡x+12ln⁡|csc⁡x−cot⁡x|+C
* ∫cscnaxdx=−cscn−2axcot⁡axa(n−1)+n−2n−1∫cscn−2axdx (for n≠1)
* ∫dxcsc⁡x+1=x−2cot⁡x2+1+C
* ∫dxcsc⁡x−1=−x+2cot⁡x2−1+C

## Integrands involving only cotangent

* ∫cot⁡axdx=1aln⁡|sin⁡ax|+C
* ∫cot2xdx=−cot⁡x−x+C
* ∫cotnaxdx=−1a(n−1)cotn−1ax−∫cotn−2axdx(for n≠1)
* ∫dx1+cot⁡ax=∫tan⁡axdxtan⁡ax+1=x2−12aln⁡|sin⁡ax+cos⁡ax|+C
* ∫dx1−cot⁡ax=∫tan⁡axdxtan⁡ax−1=x2+12aln⁡|sin⁡ax−cos⁡ax|+C

## Integrands involving both sine and cosine

An integral that is a rational function of the sine and cosine can be evaluated using Bioche's rules.

* ∫dxcos⁡ax±sin⁡ax=1a2ln⁡|tan⁡(ax2±π8)|+C
* ∫dx(cos⁡ax±sin⁡ax)2=12atan⁡(ax∓π4)+C
* ∫dx(cos⁡x+sin⁡x)n=12(n−1)(sin⁡x−cos⁡x(cos⁡x+sin⁡x)n−1+(n−2)∫dx(cos⁡x+sin⁡x)n−2)
* ∫cos⁡axdxcos⁡ax+sin⁡ax=x2+12aln⁡|sin⁡ax+cos⁡ax|+C
* ∫cos⁡axdxcos⁡ax−sin⁡ax=x2−12aln⁡|sin⁡ax−cos⁡ax|+C
* ∫sin⁡axdxcos⁡ax+sin⁡ax=x2−12aln⁡|sin⁡ax+cos⁡ax|+C
* ∫sin⁡axdxcos⁡ax−sin⁡ax=−x2−12aln⁡|sin⁡ax−cos⁡ax|+C
* ∫cos⁡axdx(sin⁡ax)(1+cos⁡ax)=−14atan2ax2+12aln⁡|tan⁡ax2|+C
* ∫cos⁡axdx(sin⁡ax)(1−cos⁡ax)=−14acot2ax2−12aln⁡|tan⁡ax2|+C
* ∫sin⁡axdx(cos⁡ax)(1+sin⁡ax)=14acot2(ax2+π4)+12aln⁡|tan⁡(ax2+π4)|+C
* ∫sin⁡axdx(cos⁡ax)(1−sin⁡ax)=14atan2(ax2+π4)−12aln⁡|tan⁡(ax2+π4)|+C
* ∫(sin⁡ax)(cos⁡ax)dx=12asin2ax+C
* ∫(sin⁡a1x)(cos⁡a2x)dx=−cos⁡((a1−a2)x)2(a1−a2)−cos⁡((a1+a2)x)2(a1+a2)+C(for |a1|≠|a2|)
* ∫(sinnax)(cos⁡ax)dx=1a(n+1)sinn+1ax+C(for n≠−1)
* ∫(sin⁡ax)(cosnax)dx=−1a(n+1)cosn+1ax+C(for n≠−1)
* ∫(sinnax)(cosmax)dx=−(sinn−1ax)(cosm+1ax)a(n+m)+n−1n+m∫(sinn−2ax)(cosmax)dx(for m,n>0)=(sinn+1ax)(cosm−1ax)a(n+m)+m−1n+m∫(sinnax)(cosm−2ax)dx(for m,n>0)
* ∫dx(sin⁡ax)(cos⁡ax)=1aln⁡|tan⁡ax|+C
* ∫dx(sin⁡ax)(cosnax)=1a(n−1)cosn−1ax+∫dx(sin⁡ax)(cosn−2ax)(for n≠1)
* ∫dx(sinnax)(cos⁡ax)=−1a(n−1)sinn−1ax+∫dx(sinn−2ax)(cos⁡ax)(for n≠1)
* ∫sin⁡axdxcosnax=1a(n−1)cosn−1ax+C(for n≠1)
* ∫sin2axdxcos⁡ax=−1asin⁡ax+1aln⁡|tan⁡(π4+ax2)|+C
* ∫sin2axdxcosnax=sin⁡axa(n−1)cosn−1ax−1n−1∫dxcosn−2ax(for n≠1)
* ∫sin2x1+cos2xdx=2arctangant⁡(tan⁡x2)−x(for x in]−π2;+π2[)=2arctangant⁡(tan⁡x2)−arctangant⁡(tan⁡x)(this time x being any real number )
* ∫sinnaxdxcos⁡ax=−sinn−1axa(n−1)+∫sinn−2axdxcos⁡ax(for n≠1)
* ∫sinnaxdxcosmax={sinn+1axa(m−1)cosm−1ax−n−m+2m−1∫sinnaxdxcosm−2ax(for m≠1)sinn−1axa(m−1)cosm−1ax−n−1m−1∫sinn−2axdxcosm−2ax(for m≠1)−sinn−1axa(n−m)cosm−1ax+n−1n−m∫sinn−2axdxcosmax(for m≠n)
* ∫cos⁡axdxsinnax=−1a(n−1)sinn−1ax+C(for n≠1)
* ∫cos2axdxsin⁡ax=1a(cos⁡ax+ln⁡|tan⁡ax2|)+C
* ∫cos2axdxsinnax=−1n−1(cos⁡axasinn−1ax+∫dxsinn−2ax)(for n≠1)
* ∫cosnaxdxsinmax={−cosn+1axa(m−1)sinm−1ax−n−m+2m−1∫cosnaxdxsinm−2ax(for n≠1)−cosn−1axa(m−1)sinm−1ax−n−1m−1∫cosn−2axdxsinm−2ax(for m≠1)cosn−1axa(n−m)sinm−1ax+n−1n−m∫cosn−2axdxsinmax(for m≠n)

## Integrands involving both sine and tangent

* ∫(sin⁡ax)(tan⁡ax)dx=1a(ln⁡|sec⁡ax+tan⁡ax|−sin⁡ax)+C
* ∫tannaxdxsin2ax=1a(n−1)tann−1(ax)+C(for n≠1)

## Integrand involving both cosine and tangent

* ∫tannaxdxcos2ax=1a(n+1)tann+1ax+C(for n≠−1)

## Integrand involving both sine and cotangent

* ∫cotnaxdxsin2ax=−1a(n+1)cotn+1ax+C(for n≠−1)

## Integrand involving both cosine and cotangent

* ∫cotnaxdxcos2ax=1a(1−n)tan1−nax+C(for n≠1)

## Integrand involving both secant and tangent

* ∫(sec⁡x)(tan⁡x)dx=sec⁡x+C

## Integrand involving both cosecant and cotangent

* ∫(csc⁡x)(cot⁡x)dx=−csc⁡x+C

## Integrals in a quarter period

Using the beta function B(a,b) one can write

* ∫0π2sinnxdx=∫0π2cosnxdx=12B(n+12,12)={n−1n⋅n−3n−2⋯34⋅12⋅π2,if n is evenn−1n⋅n−3n−2⋯45⋅23,if n is odd and more than 11,if n=1

Using the modified Struve functions Lα(x) and modified Bessel functions Iα(x) one can write

* ∫0π2exp⁡(k⋅sin⁡(x))dx=π2(I0(k)+L0(k))

## Integrals with symmetric limits

* ∫−ccsin⁡xdx=0
* ∫−cccos⁡xdx=2∫0ccos⁡xdx=2∫−c0cos⁡xdx=2sin⁡c
* ∫−cctan⁡xdx=0
* ∫−a2a2x2cos2nπxadx=a3(n2π2−6)24n2π2(for n=1,3,5...)
* ∫−a2a2x2sin2nπxadx=a3(n2π2−6(−1)n)24n2π2=a324(1−6(−1)nn2π2)(for n=1,2,3,...)

## Integral over a full circle

* ∫02πsin2m+1xcosnxdx=0n,m∈ℤ
* ∫02πsinmxcos2n+1xdx=0n,m∈ℤ

## See also

* Trigonometric integral

## References

1. ↑ Bresock, Krista (2022-01-01). "Student Understanding of the Definite Integral When Solving Calculus Volume Problems". Graduate Theses, Dissertations, and Problem Reports. doi:10.33915/etd.11491. https://researchrepository.wvu.edu/etd/11491.

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Lists of integrals

* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

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